Calculus Examples

$f (x) = \frac{1}{4} x^{4} + \frac{1}{4} + \frac{1}{3} x^{3}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} + \frac{1}{4} + \frac{1}{3} x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$ .

$\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{3} + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{4 x^{3}}{4} + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.2.5

Cancel the common factor of $4$ .

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Step 1.2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.2.5.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d x} [\frac{1}{4}] + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.3

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4}$ with respect to $x$ is $0$ .

$x^{3} + 0 + \frac{d}{d x} [\frac{1}{3} x^{3}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 1.4.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$x^{3} + 0 + \frac{1}{3} \frac{d}{d x} [x^{3}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} + 0 + \frac{1}{3} (3 x^{2})$

Step 1.4.3

Combine $3$ and $\frac{1}{3}$ .

$x^{3} + 0 + \frac{3}{3} x^{2}$

Step 1.4.4

Combine $\frac{3}{3}$ and $x^{2}$ .

$x^{3} + 0 + \frac{3 x^{2}}{3}$

Step 1.4.5

Cancel the common factor of $3$ .

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Step 1.4.5.1

Cancel the common factor.

$x^{3} + 0 + \frac{3 x^{2}}{3}$

Step 1.4.5.2

Divide $x^{2}$ by $1$ .

$x^{3} + 0 + x^{2}$

Step 1.5

Add $x^{3}$ and $0$ .

$f' (x) = x^{3} + x^{2}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [x^{2}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 3 x^{2} + 2 x$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 2 x$ .

$3 x^{2} + 2 x$